{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T13:34:28Z","timestamp":1773840868280,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T00:00:00Z","timestamp":1747958400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T00:00:00Z","timestamp":1747958400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["2110780"],"award-info":[{"award-number":["2110780"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s10898-025-01504-6","type":"journal-article","created":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T01:52:12Z","timestamp":1747965132000},"page":"509-534","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Distributionally Robust Optimization with Polynomial Robust Constraints"],"prefix":"10.1007","volume":"92","author":[{"given":"Jiawang","family":"Nie","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6658-151X","authenticated-orcid":false,"given":"Suhan","family":"Zhong","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,23]]},"reference":[{"issue":"1","key":"1504_CR1","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1137\/18M1172843","volume":"29","author":"AA Ahmadi","year":"2019","unstructured":"Ahmadi, A.A., De Klerk, E., Hall, G.: Polynomial norms. SIAM J. Optim. 29(1), 399\u2013422 (2019)","journal-title":"SIAM J. Optim."},{"key":"1504_CR2","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/s10107-011-0499-2","volume":"137","author":"AA Ahmadi","year":"2013","unstructured":"Ahmadi, A.A., Olshevsky, A., Parrilo, P.A., Tsitsiklis, J.N.: NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Program. 137, 453\u2013476 (2013)","journal-title":"Math. Program."},{"issue":"2","key":"1504_CR3","doi-asserted-by":"publisher","first-page":"811","DOI":"10.1137\/110856010","volume":"23","author":"AA Ahmadi","year":"2013","unstructured":"Ahmadi, A.A., Parrilo, P.: A complete characterization of the gap between convexity and sos-convexity. SIAM J. Optim. 23(2), 811\u2013833 (2013)","journal-title":"SIAM J. Optim."},{"key":"1504_CR4","volume-title":"Lectures on modern convex optimization","author":"A Ben-Tal","year":"2012","unstructured":"Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization. SIAM, Philadelphia, PA (2012)"},{"issue":"2","key":"1504_CR5","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1287\/mnsc.2017.2952","volume":"65","author":"D Bertsimas","year":"2019","unstructured":"Bertsimas, D., Sim, M., Zhang, M.: Adaptive distributionally robust optimization. Manag. Sci. 65(2), 604\u2013618 (2019)","journal-title":"Manag. Sci."},{"issue":"1","key":"1504_CR6","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s10589-020-00234-7","volume":"78","author":"Y Chen","year":"2021","unstructured":"Chen, Y., Sun, H., Xu, H.: Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems. Comput. Optim. Appl. 78(1), 205\u2013238 (2021)","journal-title":"Comput. Optim. Appl."},{"key":"1504_CR7","unstructured":"H.\u00a0T.M.\u00a0Chu, K-C.\u00a0Toh and Y.\u00a0Zhang, On regularized square-root regression problems: distributionally robust interpretation and fast computations, Preprint, 2021. arXiv:2109.03632"},{"issue":"3","key":"1504_CR8","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1287\/opre.1090.0741","volume":"58","author":"E Delage","year":"2010","unstructured":"Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595\u2013612 (2010)","journal-title":"Oper. Res."},{"key":"1504_CR9","doi-asserted-by":"crossref","unstructured":"C.\u00a0A.\u00a0Garc\u00eda Trillos and N.\u00a0Garc\u00eda Trillos, On the regularized risk of distributionally robust learning over deep neural networks, Res. Math. Sci., 9(3), Paper No. 54, pp 32, 2022","DOI":"10.1007\/s40687-022-00349-9"},{"issue":"4","key":"1504_CR10","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1287\/opre.1090.0795","volume":"58","author":"J Goh","year":"2010","unstructured":"Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4), 902\u2013917 (2010)","journal-title":"Oper. Res."},{"issue":"3","key":"1504_CR11","doi-asserted-by":"publisher","first-page":"1014","DOI":"10.1007\/s10957-022-02063-6","volume":"194","author":"M G\u00fcrb\u00fczbalaban","year":"2022","unstructured":"G\u00fcrb\u00fczbalaban, M., Ruszczy\u0144ski, A., Zhu, L.: A stochastic subgradient method for distributionally robust non-convex learning. J. Optim. Theory Appl. 194(3), 1014\u20131041 (2022)","journal-title":"J. Optim. Theory Appl."},{"issue":"4\u20135","key":"1504_CR12","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1080\/10556780802699201","volume":"24","author":"D Henrion","year":"2009","unstructured":"Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4\u20135), 761\u2013779 (2009)","journal-title":"Optim. Methods Softw."},{"key":"1504_CR13","doi-asserted-by":"publisher","DOI":"10.1142\/q0252","volume-title":"The Moment-SOS Hierarchy","author":"D Henrion","year":"2020","unstructured":"Henrion, D., Korda, M., Lasserre, J.: The Moment-SOS Hierarchy. World Scientific, Singapore (2020)"},{"key":"1504_CR14","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1090\/S0002-9939-07-09068-5","volume":"136","author":"C Hillar","year":"2008","unstructured":"Hillar, C., Nie, J.: An elementary and constructive proof of Hilbert\u2019s 17th Problem for matrices. Proceedings of the AMS 136, 73\u201376 (2008)","journal-title":"Proceedings of the AMS"},{"key":"1504_CR15","doi-asserted-by":"crossref","unstructured":"E.\u00a0de. Klerk, D.\u00a0Kuhn, and K.\u00a0Postek, Distributionally robust optimization with polynomial densities: theory, models and algorithms, Math. Program., 181, 265\u2013296, 2020","DOI":"10.1007\/s10107-019-01429-5"},{"key":"1504_CR16","doi-asserted-by":"publisher","first-page":"1995","DOI":"10.1137\/080728214","volume":"19","author":"JB Lasserre","year":"2009","unstructured":"Lasserre, J.B.: Convexity in semi-algebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995\u20132014 (2009)","journal-title":"SIAM J. Optim."},{"key":"1504_CR17","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107447226","volume-title":"Introduction to Polynomial and Semi-algebraic Optimization","author":"JB Lasserre","year":"2015","unstructured":"Lasserre, J.B.: Introduction to Polynomial and Semi-algebraic Optimization. Cambridge University Press, Cambridge (2015)"},{"issue":"1","key":"1504_CR18","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1007\/s10107-019-01434-8","volume":"185","author":"JB Lasserre","year":"2021","unstructured":"Lasserre, J.B., Weisser, T.: Distributionally robust polynomial chance-constraints under mixture ambiguity sets. Math. Program. 185(1), 409\u2013453 (2021)","journal-title":"Math. Program."},{"key":"1504_CR19","doi-asserted-by":"crossref","unstructured":"Laurent, M.: Sums of squares, moment matrices and optimization over polynomials, Emerging Applications of Algebraic Geometry of IMA Volumes in Mathematics and its Applications 149, pp. 157\u2013270, Springer, (2009)","DOI":"10.1007\/978-0-387-09686-5_7"},{"issue":"1","key":"1504_CR20","doi-asserted-by":"publisher","first-page":"159","DOI":"10.3934\/naco.2021057","volume":"12","author":"F Lin","year":"2022","unstructured":"Lin, F., Fang, X.: Distributionally robust optimization: a review on theory and applications. Numer. Algebra Control Optim. 12(1), 159\u2013212 (2022)","journal-title":"Numer. Algebra Control Optim."},{"key":"1504_CR21","unstructured":"Lofberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB, IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508), IEEE, (2004)"},{"key":"1504_CR22","first-page":"37","volume":"26","author":"M Mevissen","year":"2013","unstructured":"Mevissen, M., Ragnoli, E., Yu, J.: Data-driven distributionally robust polynomial optimization. Advances in NIPS 26, 37\u201345 (2013)","journal-title":"Advances in NIPS"},{"issue":"3","key":"1504_CR23","doi-asserted-by":"publisher","first-page":"1996","DOI":"10.1137\/19M1263121","volume":"30","author":"J Milz","year":"2020","unstructured":"Milz, J., Ulbrich, M.: An approximation scheme for distributionally robust nonlinear optimization. SIAM J. Optim. 30(3), 1996\u20132025 (2020)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"1504_CR24","doi-asserted-by":"publisher","first-page":"398","DOI":"10.1287\/moor.1110.0498","volume":"36","author":"J Nie","year":"2011","unstructured":"Nie, J.: Polynomial matrix inequality and semidefinite representation. Math. Oper. Res. 36(3), 398\u2013415 (2011)","journal-title":"Math. Oper. Res."},{"key":"1504_CR25","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/s11464-012-0187-4","volume":"7","author":"J Nie","year":"2012","unstructured":"Nie, J.: Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces. Front. Math. China 7, 321\u2013346 (2012)","journal-title":"Front. Math. China"},{"issue":"1\u20132","key":"1504_CR26","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s10107-012-0589-9","volume":"142","author":"J Nie","year":"2013","unstructured":"Nie, J.: Certifying convergence of Lasserre\u2019s hierarchy via flat truncation. Math. Program. 142(1\u20132), 485\u2013510 (2013)","journal-title":"Math. Program."},{"issue":"6","key":"1504_CR27","doi-asserted-by":"publisher","first-page":"1243","DOI":"10.1007\/s10208-014-9225-9","volume":"14","author":"J Nie","year":"2014","unstructured":"Nie, J.: The $$\\cal{A} $$-Truncated $$K$$-Moment Problem. Found. Comput. Math. 14(6), 1243\u20131276 (2014)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"1504_CR28","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/16M1083384","volume":"1","author":"J Nie","year":"2017","unstructured":"Nie, J.: Symmetric tensor nuclear norms. SIAM J. Appl. Algebra Geometry 1(1), 599\u2013625 (2017)","journal-title":"SIAM J. Appl. Algebra Geometry"},{"issue":"1","key":"1504_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10589-017-9973-y","volume":"70","author":"J Nie","year":"2018","unstructured":"Nie, J., Zhang, X.: Real eigenvalues of nonsymmetric tensors. Comp. Opt. and Appl. 70(1), 1\u201332 (2018)","journal-title":"Comp. Opt. and Appl."},{"issue":"1\u20132","key":"1504_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10107-018-1276-2","volume":"178","author":"J Nie","year":"2019","unstructured":"Nie, J.: Tight relaxations for polynomial optimization and lagrange multiplier expressions. Math. Program. 178(1\u20132), 1\u201337 (2019)","journal-title":"Math. Program."},{"key":"1504_CR31","doi-asserted-by":"crossref","unstructured":"J.\u00a0Nie, Moment and Polynomial Optimization, SIAM, 2023","DOI":"10.1137\/1.9781611977608"},{"issue":"2","key":"1504_CR32","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1080\/10556788.2019.1649672","volume":"35","author":"J Nie","year":"2020","unstructured":"Nie, J., Yang, L., Zhong, S.: Stochastic polynomial optimization. Optim. Methods Softw. 35(2), 329\u2013347 (2020)","journal-title":"Optim. Methods Softw."},{"issue":"12","key":"1504_CR33","first-page":"1","volume":"94","author":"J Nie","year":"2023","unstructured":"Nie, J., Yang, L., Zhong, S., Zhou, G.: Distributionally robust optimization with moment ambiguity sets. J. Sci. Comput. 94(12), 1\u201327 (2023)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"1504_CR34","doi-asserted-by":"publisher","first-page":"969","DOI":"10.1512\/iumj.1993.42.42045","volume":"42","author":"M Putinar","year":"1993","unstructured":"Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969\u2013984 (1993)","journal-title":"Indiana Univ. Math. J."},{"key":"1504_CR35","doi-asserted-by":"crossref","unstructured":"A.\u00a0Shapiro, D.\u00a0Dentcheva, and A.\u00a0Ruszczy\u0144ski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, 2014","DOI":"10.1137\/1.9781611973433"},{"key":"1504_CR36","doi-asserted-by":"crossref","unstructured":"J.\u00a0Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11(1-4), 625\u2013653, 1999","DOI":"10.1080\/10556789908805766"},{"issue":"1","key":"1504_CR37","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/s10107-022-01889-2","volume":"200","author":"H Sun","year":"2023","unstructured":"Sun, H., Shapiro, A., Chen, X.: Distributionally robust stochastic variational inequalities. Math. Program. 200(1), 279\u2013317 (2023)","journal-title":"Math. Program."},{"issue":"3","key":"1504_CR38","doi-asserted-by":"publisher","first-page":"2387","DOI":"10.1287\/moor.2021.1213","volume":"47","author":"L Xin","year":"2022","unstructured":"Xin, L., Goldberg, D.A.: Distributionally robust inventory control when demand is a martingale. Math. Oper. Res. 47(3), 2387\u20132414 (2022)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"1504_CR39","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1007\/s10107-017-1143-6","volume":"169","author":"H Xu","year":"2018","unstructured":"Xu, H., Liu, Y., Sun, H.: Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods. Math. Program. 169(2), 489\u2013529 (2018)","journal-title":"Math. Program."},{"issue":"4","key":"1504_CR40","doi-asserted-by":"publisher","first-page":"3743","DOI":"10.1109\/TSG.2018.2834564","volume":"10","author":"Y Yang","year":"2018","unstructured":"Yang, Y., Wu, W.: A Distributionally Robust Optimization Model for Real-Time Power Dispatch in Distribution Networks. IEEE Transactions on Smart Grid 10(4), 3743\u20133752 (2018)","journal-title":"IEEE Transactions on Smart Grid"},{"key":"1504_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/s40305-023-00519-8","author":"L Yang","year":"2023","unstructured":"Yang, L., Yang, Y., Zhong, S.: Global optimization for the portfolio selection model with high-order moments. J. Oper. Res. Soc. China (2023). https:\/\/doi.org\/10.1007\/s40305-023-00519-8","journal-title":"J. Oper. Res. Soc. China"},{"issue":"3","key":"1504_CR42","doi-asserted-by":"publisher","first-page":"1855","DOI":"10.1137\/15M1038529","volume":"26","author":"J Zhang","year":"2016","unstructured":"Zhang, J., Xu, H., Zhang, L.: Quantitative stability analysis for distributionally robust optimization with moment constraints. SIAM J. Optim. 26(3), 1855\u20131862 (2016)","journal-title":"SIAM J. Optim."},{"issue":"5","key":"1504_CR43","doi-asserted-by":"publisher","first-page":"1155","DOI":"10.1287\/opre.1080.0684","volume":"57","author":"S Zhu","year":"2009","unstructured":"Zhu, S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5), 1155\u20131168 (2009)","journal-title":"Oper. Res."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-025-01504-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-025-01504-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-025-01504-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T06:49:39Z","timestamp":1751611779000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-025-01504-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,23]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["1504"],"URL":"https:\/\/doi.org\/10.1007\/s10898-025-01504-6","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,23]]},"assertion":[{"value":"29 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 May 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 May 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest\/Competing interests"}}]}}